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I'm a bit lost at the results part of my thesis. I have conducted an generalized maximum likelihood (GML) univariate test with Y variable: dependent variable one X variable: independent (consists of 2 conditions)

I have a lot of covariates to control for. So i performed different ANOVA's with just one covariate at a time.

Some covariates turn out to be significant AND the X variable is also significant.
Some covariates turn out to be significant but the X variable is not significant.

The problem:
I previously wrote in the results section of my thesis, that when a covariate turns out to be significant and the X was less significant than in the analysis without the covariates, this means I found a main effect but it did not change the interpretation of the results.

But what is the case if the some covariates are not significant, but the x variable is significant and even more significant than in the analysis without covariates?

What can I conclude from this finding? I tried to google it but i couldn't find it. I hope you can help me.

A lot of thanks in advance.

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Some idea. Hopefully you'll get a few responses and all together there will be something useful.

(1) I usually use the term "main effects" to refer to a multivariate regression model before I've addressed interactions, or that part of a multivariate model that includes the variables but not the interactions. Thus in y ~ x + z + x*z, x and z are the main effects.

(2) It sounds like you're doing bivariate analysis to look for confounders that should be included in a final multivariate regression model. This isn't the best way to build a regression model. I'm not sure there is a best way, and it does depend on what you're trying to achieve with your model. If you're just trying to find the best estimate of your main variable of interest then include all/most variables, if you interested in a group of factors that are all together associated with the outcome some model building is needed.

(3) Some covariates turn out to be significant AND the X variable is also significant = the covariate is associated with the outcome, but this does not say anything about whether the covariate is associated with x. If it is associated with outcome and x it could be a confounder

(4) But what is the case if the some covariates are not significant, but the x variable is significant and even more significant than in the analysis without covariates? = is this the classic "suppressor variable" ? (http://www.uvm.edu/~dhowell/gradstat/psych341/lectures/MultipleRegression/multreg3.html) Here I'm assuming you mean statistically significant.

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